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Algorithm and their Computation

M491/M691: Algorithms and their Computations

No run this semester


Overview

This course will enroll a combination of mathematically mature undergraduates and first or second year graduate students working in groups on a set of established problems. These problems will be sufficiently elementary to allow progress to made reasonably quickly to a first stage yet have enough complexity and overtones to take the more able and interested students into a research topic proper.

The common theme will be as the title suggests algorithms and their computation or experimental mathematics using computation.

There will be very few lectures beyond the first week or two and even these will be to set up the scope of projects and give background information, but in only a few cases necessary technical lectures. Learning will be through discovering about needed tools as the projects progress. In short, not much different from what is the common approach for a Research Experience for Undergraduate (REU) program or the current in-house undergraduate research course.

In the Summer, this is a regular 6-week (3 credits) class and will run the first term. In the Fall, this is a regular 3 credits class.

Limited amount of funding is available to support graduate and undergraduate students attending to the class.

Our motivations are several fold.

Scope and underlying Themes

  • There will be a focus on algorithms; their scope and proper implementation. This will require gaining a basic understanding for the numerical analysis involved as well as for the underlying mathematics. However, this is not a class about coding algorithms.
  • The idea of testing or formulating mathematical conjectures by computing a range of specific examples will be a central theme.
  • Although not a strict requirement, programming and benchmarking will be important aspects. The students could be asked to numerically explore the validity of mathematical theorem outside their scope or to test numerically the credibility of conjectures. For this reason the groups formed will have a well-chosen range of abilities in both the theory and implementation.
  • The problem certainly need not be tied to any specific application problem but we want to avoid problems contrived for their own sake and without a broader vision.

Course Topics for Summer 2015 / Fall 2015 Terms

Depending on the interest, some of the following projects will be offered.
  • Problems based on integral equations. At one level integral equations are the natural inverse of differential equations; Newton showed us this. However, they offer not just a mirror formulation but a perspective that allows transparent computations as well as leading into operators that allow a greater generality than merely being tied to differential operators. While the applications run the entire gamut of the sciences, such a general tool is not without its complexities. These provide the perfect opportunity for investigation and conjecture-formimg though computation and the topic will seek to exploit this fully. This is a very multi-level topic being assessible with only relativel minor prequisites yet easily leading into complex questions in (both linear and nonlinear) functional analysis.
    Prerequisite for mathematics undergraduates: M304, M308 and M409. In addition M412 and M417 would be useful.
  • Computational methods for geometric PDEs. Deformable domains are ubiquitous in several areas of research. We intend to focus on the characterization of equilibrium shapes via a short and basic excursion in the fields of calculus of variations and differential geometry. The emphasis will be on the more tractable setting of two dimensional deformable curves yet already containing the principal difficulties. The students will be confronted to problematics related to the effect of surface tension and more complex constraints, where intuition could be misleading.
    Prerequisite for mathematics undergraduates: M304, M308 and M409. In addition M412 and M417 would be useful.
  • Computational problems in harmonic analysis. Specifically, we will take the case of image processing although extending beyond this has considerable scope. One possible line is image de-blurring which leads directly to more general questions of regularization issues for inverting first kind integral equations or indeed even more generally of inverting compact operators with stability constraints.
    Prerequisite for mathematics undergraduates: M304, M308 and M409. In addition M412 and M417 would be useful.
  • Computational number theory. This a vast topic that at one level is very practical due to the fundamental relationship with modern cryptography. Probably no other subject uses computation to hone conjectures as much as number theory The practical ramifications make this well-suited to non-academic job opportunities while on the other hand the rich theoretical underpinning offers many opportunities for conjecture testing by computational means.
    Prerequisite for mathematics undergraduates: M304, M308 and M409. In addition one of M415, M433 or M427 would be useful.
  • High dimensional approximation. The numerical solution of many scientific problems corresponds to the approximation of a function depending on a large number of variables. According to the classical theory, the approximability of a function drastically deteriorates with the number of variables. We start with basic approximation theory questions for one parameter functions to discover what it takes to construct "the best possible approximation". We will then observe the limitations of standard techniques when the function depends on a large number of parameters and explore new theories and algorithms defeating this curse of dimensionality.
    Prerequisite for mathematics undergraduates: M221.

Summer 2015 Term

More information for the Summer 2015 term is available on the class website.

Fall 2015 Term

More information will be avalaible soon.

Prerequisite FAQs - Can I take this Class?

Student status Requirement
Mathematics undergraduate see topics descriptions.
Physics/Engineering student same as mathematics undergraduate.
Mathematics graduate student Being a graduate student
Freshman with a score of 25 on the Putnam exam Of course, you are eligible!

Contact Information

For further details interested students are encouraged to contact either